Prove that if $X$ and $Y$ are Normal and independent random variables, $X+Y$ and $X-Y$ are independent
HINT:
- Independent Gussian random variables make a Gaussian random vector.
- Affine transform $Y=A X + b$ of Gaussian random vector $X$ is Gaussian.
- Distribution of Gaussian random vector is determined by its mean vector, and covariance matrix.
- If components $X_i$ and $X_j$ of the Gaussian random vector are independent, then $\mathbb{Cov}(X_i, X_j) = 0$.
Combining facts given above, it follows that evaluation of $\mathbb{Cov}(X+Y,X-Y)$ will help establish the result needed.